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Spectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: Comparison with sinc methods

TitleSpectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: Comparison with sinc methods
Publication TypeJournal Article
Year of Publication2006
AuthorsLo, J, Shizgal, BD
JournalJournal of Chemical Physics
Volume125
Pagination17
Date PublishedNov
Type of ArticleArticle
ISBN Number0021-9606
KeywordsBOLTZMANN COLLISION OPERATOR, DISCRETE-VARIABLE REPRESENTATIONS, EIGENVALUE PROBLEMS, ELECTRON THERMALIZATION, LANCZOS METHOD, MATRIX-ELEMENTS, NONCLASSICAL ORTHOGONAL POLYNOMIALS, ORDINATE METHOD, SINGULAR CONVOLUTION, SUPERSYMMETRIC QUANTUM-MECHANICS
Abstract

Spectral methods based on nonclassical polynomials and Fourier basis functions or sinc interpolation techniques are compared for several eigenvalue problems for the Fokker-Planck and Schrodinger equations. A very rapid spectral convergence of the eigenvalues versus the number of quadrature points is obtained with the quadrature discretization method (QDM) and the appropriate choice of the weight function. The QDM is a pseudospectral method and the rate of convergence is compared with the sinc method reported by Wei [J. Chem. Phys., 110, 8930 (1999)]. In general, sinc methods based on Fourier basis functions with a uniform grid provide a much slower convergence. The paper considers Fokker-Planck equations (and analogous Schrodinger equations) for the thermalization of electrons in atomic moderators and for a quartic potential employed to model chemical reactions. The solution of the Schrodinger equation for the vibrational states of I-2 with a Morse potential is also considered. (c) 2006 American Institute of Physics.

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